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Number Systems

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Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No Hexa- decimal 160, 1, … 9, A, B, … F No

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Quantities/Counting (1 of 3) DecimalBinaryOctal Hexa- decimal 0000 1111 21022 31133 410044 510155 611066 711177 p. 33

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Quantities/Counting (2 of 3) DecimalBinaryOctal Hexa- decimal 81000108 91001119 10101012A 11101113B 12110014C 13110115D 14111016E 15111117F

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Quantities/Counting (3 of 3) DecimalBinaryOctal Hexa- decimal 16100002010 17100012111 18100102212 19100112313 20101002414 21101012515 22101102616 23101112717 Etc.

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Conversion Among Bases The possibilities: Hexadecimal DecimalOctal Binary pp. 40-46

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Quick Example 25 10 = 11001 2 = 31 8 = 19 16 Base

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Decimal to Decimal (just for fun) Hexadecimal DecimalOctal Binary Next slide…

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125 10 =>5 x 10 0 = 5 2 x 10 1 = 20 1 x 10 2 = 100 125 Base Weight

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Binary to Decimal Hexadecimal DecimalOctal Binary

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Binary to Decimal Technique –Multiply each bit by 2 n, where n is the weight of the bit –The weight is the position of the bit, starting from 0 on the right –Add the results

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Example 101011 2 => 1 x 2 0 = 1 1 x 2 1 = 2 0 x 2 2 = 0 1 x 2 3 = 8 0 x 2 4 = 0 1 x 2 5 = 32 43 10 Bit 0

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Octal to Decimal Hexadecimal DecimalOctal Binary

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Octal to Decimal Technique –Multiply each bit by 8 n, where n is the weight of the bit –The weight is the position of the bit, starting from 0 on the right –Add the results

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Example 724 8 => 4 x 8 0 = 4 2 x 8 1 = 16 7 x 8 2 = 448 468 10

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Hexadecimal to Decimal Hexadecimal DecimalOctal Binary

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Hexadecimal to Decimal Technique –Multiply each bit by 16 n, where n is the weight of the bit –The weight is the position of the bit, starting from 0 on the right –Add the results

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Example ABC 16 =>C x 16 0 = 12 x 1 = 12 B x 16 1 = 11 x 16 = 176 A x 16 2 = 10 x 256 = 2560 2748 10

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Decimal to Binary Hexadecimal DecimalOctal Binary

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Decimal to Binary Technique –Divide by two, keep track of the remainder –First remainder is bit 0 (LSB, least-significant bit) –Second remainder is bit 1 –Etc.

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Example 125 10 = ? 2 2 125 62 1 2 31 0 2 15 1 2 7 1 2 3 1 2 1 1 2 0 1 125 10 = 1111101 2

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Octal to Binary Hexadecimal DecimalOctal Binary

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Octal to Binary Technique –Convert each octal digit to a 3-bit equivalent binary representation

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Example 705 8 = ? 2 7 0 5 111 000 101 705 8 = 111000101 2

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Hexadecimal to Binary Hexadecimal DecimalOctal Binary

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Hexadecimal to Binary Technique –Convert each hexadecimal digit to a 4-bit equivalent binary representation

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Example 10AF 16 = ? 2 1 0 A F 0001 0000 1010 1111 10AF 16 = 0001000010101111 2

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Decimal to Octal Hexadecimal DecimalOctal Binary

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Decimal to Octal Technique –Divide by 8 –Keep track of the remainder

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Example 1234 10 = ? 8 8 1234 154 2 8 19 2 8 2 3 8 0 2 1234 10 = 2322 8

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Decimal to Hexadecimal Hexadecimal DecimalOctal Binary

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Decimal to Hexadecimal Technique –Divide by 16 –Keep track of the remainder

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Example 1234 10 = ? 16 1234 10 = 4D2 16 16 1234 77 2 16 4 13 = D 16 0 4

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Binary to Octal Hexadecimal DecimalOctal Binary

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Binary to Octal Technique –Group bits in threes, starting on right –Convert to octal digits

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Example 1011010111 2 = ? 8 1 011 010 111 1 3 2 7 1011010111 2 = 1327 8

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Binary to Hexadecimal Hexadecimal DecimalOctal Binary

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Binary to Hexadecimal Technique –Group bits in fours, starting on right –Convert to hexadecimal digits

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Example 1010111011 2 = ? 16 10 1011 1011 2 B B 1010111011 2 = 2BB 16

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Octal to Hexadecimal Hexadecimal DecimalOctal Binary

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Octal to Hexadecimal Technique –Use binary as an intermediary

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Example 1076 8 = ? 16 1 0 7 6 001 000 111 110 2 3 E 1076 8 = 23E 16

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Hexadecimal to Octal Hexadecimal DecimalOctal Binary

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Hexadecimal to Octal Technique –Use binary as an intermediary

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Example 1F0C 16 = ? 8 1 F 0 C 0001 1111 0000 1100 1 7 4 1 4 1F0C 16 = 17414 8

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Exercise – Convert... Dont use a calculator! DecimalBinaryOctal Hexa- decimal 33 1110101 703 1AF Skip answer Answer

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Exercise – Convert … DecimalBinaryOctal Hexa- decimal 331000014121 117111010116575 4511110000117031C3 4311101011116571AF Answer

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Binary Addition (1 of 2) Two 1-bit values ABA + B 000 011 101 1110 two

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Binary Addition (2 of 2) Two n-bit values –Add individual bits –Propagate carries –E.g., 10101 21 + 11001 + 25 101110 46 11

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Multiplication (1 of 2) Binary, two 1-bit values AB A B 000 010 100 111

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Multiplication (2 of 2) Binary, two n-bit values –As with decimal values –E.g., 1110 x 1011 1110 1110 0000 1110 10011010

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